Peel the pineapple and grate fruit finely. Place in a bowl and pour the lime juice over it. Boil the sugar with ¾ pint of the water for 4 minutes, then pour over the fruit. Add the remaining water, and stir to mix well. Let it stand for 1 hour. Strain, pressing the fruit down into the sieve to extract all the juice. Serve in tumblers with crushed ice, or add 1½ to 2 oz. rum to each serving.